The Un-Unified Field: And Other Problems

By Miles Mathis

This book is the first collection of science papers by Miles Mathis. Its topics include various problems in physics and math, beginning with the famous Unified Field problem of Einstein and string theory. These problems are solved with a simplified math and clear explanations. Other problems addressed include Bode's Law, the recent Saturn Anomaly, Quantum Chromodynamics, the ellipse, and Goldbach's Conjecture.

• Language: English
• Publisher: AuthorHouse
• Release date: Jun 30, 2010
• ISBN: 9781452005157

Miles Mathis

Sometimes called the New Leonardo, Miles Mathis is a wide-ranging thinker and creator. His two websites offer the reader everything from science and math to art, poetry, and criticism. Mathis is known worldwide for his fearlessness in attacking all power structures, and no one else on the web has produced such an impressive and extended analysis of modern art and science in so short a time. Some older critics have created greater bodies of work, but Mathis is unique in that he criticizes from within, as a working scientist and artist. In this way his critiques are never abstract or academic: they are instead blisteringly specific, down to the precise line where a famous proof goes wrong.

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UNIFIED FIELDS

in DISGUISE

Both Newton’s and Coulomb’s famous equations are unified field equations in disguise. This was not understood until I pulled them apart, showing what the constant is in each equation and how it works mechanically.

A unified field equation does not need to unify all four of the presently postulated fields. To qualify for unification, it only has to unify two of them. The unified field equations that will be unmasked in this paper both unify the gravitational field with the electromagnetic field. This unification of gravity and E/M was the great project of Einstein and is now the great project of string theory. But neither Einstein nor string theory has presented a simple unified field equation. As time has passed this has seemed more and more difficult to achieve, and more and more difficult math has been brought in to attack the problem. But it turns out the answer was always out of reach because the question was wrong. We were seeking to unify fields when we should have been seeking to segregate them. We already had two unified field equations, which is why they couldn’t be unified. We were trying to rejoin a couple that was already happily married.

Yes, both Newton and Coulomb discovered unified field equations. That is why their two equations look so much alike. But the two equations unify in different ways. Newton was unaware of the E/M field as we know it now, so he did not realize that his heuristic equation contained both fields. And Coulomb was working on electrostatics, and likewise did not realize that his equation included gravity. So the E/M field is hidden inside Newton’s equation, and the gravitational field is hidden inside Coulomb’s equation.

Let’s look at Newton’s equation first.

F = GMm/r²

We have had this lovely unified field equation since 1687. But how can we get two fields when we only have mass involved? Well, we remember that Newton invented the modern idea of mass with this equation. That is to say, he pretty much invented that variable on his own. He let that variable stand for what we now call mass, but it turns out he compressed the equation a bit too much. He wanted the simplest equation possible, but in this form it is so simple it hides the mechanics of the field. It would have been better if Newton had written the equation like this:

F = G(DV)(dv)/r²

He should have written each mass as a density and a volume. Mass is not a fundamental characteristic, like density or volume is. To know a mass, you have to know both a density and a volume. But to know a volume, you only need to know lengths. Likewise with density. Density, like volume, can be measured only with a yardstick. You will say that if density and volume can be measured with a yardstick, so can mass, since mass is defined by density and volume. True. But mass is a step more abstract, since it requires both measurements. Mass requires density and volume. But density and volume do not require mass.

Once we have density and volume in Newton’s equation, we can assign density to one field and volume to the other. We let volume define the gravitational field and we let density define the E/M field. Both fields then fall off with the square of the radius, simply because each field is spherical. There is nothing mysterious about a spherical field diminishing by the inverse square law: just look at the equation for the surface area of the sphere:

S = 4πr²

Double the radius, quadruple the surface area. Or, to say the same thing, double the radius, divide the field density by 4. If a field is caused by spherical emission, then it will diminish by the inverse square law. Quite simple.

The biggest pill to swallow is the necessary implication that gravity is now dependent only on radius. If gravity is a function of volume, and no longer of density, then gravity is not a function of mass. We have separated the variables and given density to the E/M field, so gravity is no longer a function of density. If gravity is a function of volume alone, then with a sphere gravity is a function of radius, and nothing else.

It is only the compound or unified field that is a function of mass. Yes, Newton’s equation still works like it always did, and in that equation the total force field is a function of mass. But in my separated field, solo gravity is not a function of mass. It is a function of radius and radius alone.

Now we only need to assign density mechanically. I have given it to E/M, but what part of the E/M field does it apply to? Well, it must apply to the emission. Newton’s equation is not telling us the density of the bodies in the field, it is telling us the density of the emitted field. Of course, one is a function of the other. If you have a denser Moon, it will emit a denser E/M field. But, as a matter of mechanics, the variable D applies to the density of the emitted field. It is the density of photons emitted by the matter creating the unified field.

Finally, what is G, in this analysis? G is the transform between the two fields. It is a sort of scaling constant. As we have seen, one field—solo gravity—is determined by the radius of a macro-object, like a Moon or planet or a marble. The other field is determined by the density of emitted photons. But these two fields are not operating on the same scale. To put both fields into the same equation, we must scale one field to the other. We are using both fields to find a unified force, so we must discover how force is transmitted in each field. In the E/M field, force is transmitted by the direct contact of the photons. That is, the force is felt at that level. It can be measured from any level of size, but it is being transmitted at the level of the photon. But since gravity is now a function of volume alone, it is not a function of photon size or energy. It is a function of matter itself; that is, of the atoms that make up matter. Therefore, G is a scaling constant between atoms and photons. To say it another way, G is taking the volume down to the level of size of the density, so that they may be multiplied together to find a force. Without that scaling constant, the volume would be way too large to combine directly to the density, and we would get the wrong force. By this analysis, we may assume that the photon involved in E/M transmission is about G times the proton, in size.

Now we continue on to Coulomb’s equation:

F = kq1q2/r²

One hundred years after Newton, we got another unified field equation. Here we have charges instead of masses, and the constant is different, but otherwise the equation looks the same as Newton’s. Physicists have always wondered why the equations are so similar, but until now, no one really knew. No one understood that they are both the same equation, in a different disguise.

Coulomb’s constant is another scaling constant, like G. Instead of scaling smaller, like G, k scales larger. Coulomb’s constant takes us up from the Bohr radius to the radius of macro-objects like Coulomb’s pith balls. It turns the single electron charge into a field charge.

But where is the gravitational field in Coulomb’s equation? If we study charge, we find that it has the same fundamental dimensions as mass. The statcoulomb* has dimensions of M¹/² L³/² T -1. [Mass, Length, Time]. This gives the total charge of two particles the cgs dimension ML³/T² . But mass has the dimensions L³/T²,** which makes the total charge M². So we can treat Coulomb’s charges just like Newton’s masses. We write the equation like this:

F = k(DV)(dv)/r²

Once again, the volume is the gravitational field and the density is the E/M field. The single electron is in the emitted field of the nucleus, and D gives us the density of that field. But this time the expressed field is the E/M field and the hidden field is gravity. So we have to scale the electromagnetic field up to the unified field we are measuring with our instruments.

If k and G had been the same number, all this would have been seen earlier. It would have then been easy to see that Coulomb’s equation was just the inverse of Newton’s equation. But because the constants were not the same number, the problem was hidden.

In scaling up and scaling down, we don’t simply reverse the scale. It is a bit more complex than that, as you have seen. In scaling down, we go from atomic size to photon size. In scaling up, we go from atomic size to our own size.

Unifying the two major fields of physics like this must have huge mathematical and theoretical consequences. Because Coulomb’s equation is a unified field equation, gravity must have a much larger part to play in quantum mechanics and quantum electrodynamics. In a later chapter, I will show that gravity is 10²² times stronger at the quantum level than the standard model believes. Gravity must also move into the field of the strong force, and require a complete overhaul there.

By the same token, the E/M field must invade general relativity, requiring a complete reassessment of the compound forces. At all levels of size, we will find both fields at work, creating a compound field in which each field is in opposition to the other.

*This analysis also works with the Coulomb and SI.

**Maxwell showed this in article 5 [chapter 1] of his Treatise

on Electricity and Magnetism. a = m/r², s = at²/2, m = 2sr²/t²

A DISPROOF of NEWTON’S

FUNDAMENTAL

LEMMAE

Newton published his Principia in 1687. Except for Einstein’s Relativity corrections, the bulk of the text has remained uncontested since then. It has been the backbone of trigonometry, calculus, and classical physics and, for the most part, still is. It is the fundamental text of kinematics, gravity, and many other subjects.

In this chapter I will show a simple and straightforward disproof of one of Newton’s first and most fundamental lemmae, a lemma that remains to this day the groundwork for calculus and trigonometry. My correction is important—despite the age of the text I am critiquing—due simply to the continuing importance of that text in modern mathematics and science. My correction clarifies the foundation of the calculus, a foundation that is, to this day, of great interest to pure mathematicians. In the past half-century prominent mathematicians like Abraham Robinson have continued to work on the foundation of the calculus (see Non-standard Analysis). Even at this late a date in history, important mathematical and analytical corrections must remain of interest, and a finding such as is contained in this paper is crucial to our understanding of the mathematics we have inherited. Nor has this correction ever been addressed in the historical modification of the calculus, by Cauchy or anyone else. Redefining the calculus based on limit considerations does nothing to affect the geometric or trigonometric analysis I will offer.

The first lemma in question here is Lemma VI, from Book I, section I (Of the Motion of Bodies). In that lemma, Newton’s provides the diagram below, where AB is the chord, AD is the tangent and ACB is the arc. He tells us that if we let B approach A, the angle BAD must ultimately vanish. In modern language, he is telling us that the angle goes to zero at the limit.

fig1.jpg

This is false for this reason: If we let B approach A, we must monitor the angle ABD, not the angle BAD. As B approaches A, the angle ABD approaches becoming a right angle. When B actually reaches A, the angle ABD will be a right angle. Therefore, the angle ABD can never be acute.

If we are taking B to A and may not go past A, then the angle ABD has a limit at 90o. Therefore, I claim we should see what happens to AD and AB at the limit of the angle ABD, instead of Newton’s limit for angle BAD. In other words, I propose that Newton has monitored the wrong angle.

[If you are having trouble visualizing the manipulation here, it is very simple: you must slide the entire line RBD toward A, keeping it straight always. This was the visualization of Newton, and I have not changed it here.]

In Lemma VII, Newton’s uses the previous lemma to show that at the limit the tangent, the arc and the chord are all equal. He proposes that because BAD goes to zero, AD and AB go to equality. I have just disproved this by showing that the angle ABD is 90o at the limit. If ABD is 90o at the limit, then ABD must always be greater than ADB. A triangle may not have two angles of 90o. Since AD is always longer than AB, the tangent must be greater than the chord, even at the limit. Please notice that if AB and AD are equal, then ABD must be less than 90o. But I have shown that ABD cannot be less than 90o.

1) If AB = AD, then the angle ABD must be less than 90o.

2) The angle ABD cannot be less than 90o.

QED: AB cannot equal AD.

In fact, this is precisely the reason that we can do calculations in Newton’s ultimate interval, or at the limit. If all the variables were either at zero or at equality, then we could not hope to calculate anything. Newton, very soon after proving these lemmae, used a versine equation at the ultimate interval, and he could not have done this if his variables had gone to zero or equality. Likewise, the calculus, no matter how derived or used, could not work at the limit if all the variables or functions were at zero or equality at the limit. In fact, the limit of ABD at 90o actually prevents the triangle or any of its parts from vanishing. The triangle gets smaller, but does not reach zero. Just as with the epsilon/delta proof, we get smaller but do not ever reach zero.

Some have tried to get clever and say that my claim that B never reaches A is like the paradoxes of Zeno. Am I claiming that Achilles never reaches the finish line? No, of course not. The diagram above is not equivalent to a simple diagram of motion. B is not moving toward A in the same way that Achilles approaches a finish line, and this has nothing to do with the curvature. It has to do with the implied time variable. If we diagram Achilles approaching a finish line, the time interval does not shrink as he nears the line. The time interval is constant. Plot Achilles’ motion on an x/t graph and you will see what I mean. All the little boxes on the t-axis are the same width. Or go out on the track field with Achilles and time him as he approaches the finish line. Your clock continues to go forward and tick at the same rate whether you see him 100 yards from the line or 1 inch from line.

But given the diagram above and the postulate let B go to A, it is understood that what we are doing is shrinking both the time interval and the arc distance. We are analyzing a shrinking interval, not calculating motion in space. Let B go to A does not mean analyze the motion of point B as it travels along a curve to point A. It means, let the arc length diminish. As the arc length diminishes, the variable t is also understood to diminish. Therefore, what I am saying when I say that B cannot reach A is that Δt cannot equal zero. You cannot logically analyze the interval all the way to zero, since you are analyzing motion and motion is defined by a non-zero interval.

The circle and the curve are both studies of motion. In this particular analysis, we are studying sub-intervals of motion. That subinterval, whether it is applied to space or time, cannot go to zero. Real space is non-zero space, and real time is non-zero time. We cannot study motion, velocity, force, action, or any other variable that is defined by x and t except by studying non-zero intervals. The ultimate interval is a non-zero interval, the infinitesimal is not zero, and the limit is not at zero.

The limit for any calculable variable is always greater than zero. By calculable I mean a true variable. For instance, the angle ABD is not a true variable in the problem above. It is a given. We don’t calculate it, since it is axiomatically 90o. It will be 90o in all similar problems, with any circles we could be given seeking a velocity at the tangent. The vector AD, however, will vary with different sized circles, since the curvature of different circles is different. In this way, only the angle ABD can be understood to go all the way to a limit. The other variables do not. Since they yield different solutions for different similar problems (bigger or smaller circles) they cannot be assumed to be at a zero-like limit. If they had gone all the way to some limit, they could not vary. A function at a limit should be like a constant, since the limit should prevent any further variance. Therefore, if a variable or function continues to vary under a variety of similar circumstances, you can be sure that it is not at its own limit or at zero. It is only dependent on a variable that is.

If AB and AD have real values at the limit, then we should be able to calculate those values. If we can do this we will have put a number on the infinitesimal. In fact, we do this all the time. Every time we find a number for a derivative, we put a real value on the infinitesimal. When we find an instantaneous velocity at any point on the circle, we have given a value to the infinitesimal. Remember that the tangent at any point on the circle stands for the velocity at that point. According to the diagram above, and all diagrams like it, the tangent stands for the velocity. That line is understood to be a vector whose length is the numerical value of the tangential velocity. It is commonly drawn with some recognizable length to make the illustration readable, but if it is an instantaneous velocity, the real length of the vector must be very small. Very small but not zero, since we actually find a non-zero solution for the derivative. The derivative expresses the tangent, so if the derivative is non-zero, the tangent must also be non-zero.

Some have said that since we can find sizable numbers for the tangential velocity, that vector cannot be very small. If we find that the velocity at that point is 5 m/s, for example, then shouldn’t the velocity vector have a length of 5? No, since by the way the diagram is drawn and defined, we are letting a length stand for a velocity. We are letting x stand for v. The t variable is not part of the diagram. It is implicit. It is ignored. If we are letting B approach A, then we are letting t get smaller. A velocity of 5 only means that the distance is 5 times larger than the time. If the time is tiny, the distance must be also.

Conclusion

My finding in this paper affects many things, both in pure mathematics and applied mathematics. I have proven, in a very direct fashion, that when applying the calculus to a curve, the variables or functions do not go to zero or to equality at the limit. This must have consequences both for General Relativity, which is tensor calculus applied to very small areas of curved space, and quantum electrodynamics, which applies the calculus in many ways, including quantum orbits and quantum coupling. QED has met with problems precisely when it tries to take the variables down to zero, requiring renormalization. My analysis implies that the variables do not physically go to zero, so that the assumption of infinite regression is no more than a conceptual error. The mathematical limit for calculable variables—whether in quantum physics or classical physics—is never zero. Only one in a set of variables goes to zero or to a zero-like limit (such as the angle 90o in the problem above). The other variables are non-zero at the limit. For QM or QED, this means that neither time nor length variables may go to zero when used in momentum or energy equations.

This is not to say that length and time must be quantized; it is only to say that in situations where energy is found empirically to be quantized, the other variables should also be expected to hit a limit above zero. Quantized equations must yield quantized variables. Space and time may well be continuous, but our findings—our measurements or calculations—cannot be. Meaning, we can imagine shrinking ourselves down and using tiny measuring rods to mark off sub-areas of quanta. But we cannot calculate subareas of quanta when one of our main variables—Energy—hits a limit above these subareas, and when all our data hits this same limit. The only way we could access these subareas with the variables we have is if we found a smaller quantum.

As I said, there has also been confusion on this point in the tensor calculus. In section 8 of Einstein’s paper on General Relativity¹, he gives volume to a set of coordinates that pick out a point or an event. He calls the volume of this point the natural volume, although he does not tell us what is natural about a point having volume. General Relativity starts [section 4] by postulating a point and time in space given by the coordinates dX1, dX2, dX3, dX4. This set of coordinates picks out an event, but it is still understood to be a point at an instant. This is clear since directly afterwards another set of functions is given of the form dx1, dx2, dx3, dx4. These, we are told, are the definite differentials between two infinitely proximate point-events. The volume of these differentials is given in equation 18 as dτ = ∫dx1dx2dx3dx4.

But we are also given the natural volume dτ0, which is the volume dX1, dX2, dX3, dX4. This natural volume gives us the equation 18a: dτ0 = √-gdτ

Then Einstein says, If √-g were to vanish at a point of the four-dimensional continuum, it would mean that at this point an infinitely small ‘natural’ volume would correspond to a finite volume in the co-ordinates. Let us assume this is never the case. Then g cannot change sign. . . . It always has a finite value.

According to my disproof above, all of this must be a misuse of the calculus, a misuse that is in no way made useful by importing tensors into the problem. In no kind of calculus can a set of functions that pick out an point-event be given a volume—natural, unnatural, or otherwise. If dX1, dX2, dX3, dX4 is a point-event in space, then it can have no volume, and equation 18a and everything that surrounds it is a ghost.

In the final analysis this is simply due to the definition of event. An event must be defined by some motion. If there is no motion, there is no event. All motion requires an interval. Even a non-event like a quantum sitting perfectly still implies motion in the four-vector field, since time will be passing. The non-event will have a time interval. Every possible event and non-event, in motion and at rest, requires an interval. Being at rest requires a time interval and motion requires both time and distance intervals. Therefore the event is completely determined by intervals. Not coordinates, intervals. The point and instant are not events. They are only event boundaries, boundaries that are impossible to draw with absolute precision. The instant and point are the beginning and end of an interval, but they are abstractions and estimates, not physical entities or precise spatial coordinates.

Some will answer that I have just made an apology for Einstein, saving him from my own critique. After all, he gives a theoretical interval to the point. The function dX is in the form of a differential itself, which would give it a possible extension. He may call it a point, but he dresses it as a differential. True, but he does not allow it to act like a differential, as I just showed. He disallows it from corresponding to (part of) a finite volume, since this would ruin his math. He does not allow √-g to vanish, which keeps the natural volume from invading curved space.

Newer versions of this same Riemann space have not solved this confusion, which is one of the main reasons why General Relativity still resists being incorporated into QED. Contemporary physics still believes in the point-event, the point as a physical entity (see the singularity) and the reality of the instant. All of these false notions go back to a misunderstanding of the calculus. Cauchy’s more rigorous foundation of the calculus, using the limit, the function, and the derivative, should have cleared up this confusion, but it only buried it. The problem was assumed solved since it was put more thoroughly out of sight. But it was not solved. The calculus is routinely misused in fundamental ways to this day, even (I might say especially) in the highest fields and by the biggest names.

¹Annalen der Physik, 35, 1911

ANGULAR VELOCITY

and

ANGULAR MOMENTUM

One of the greatest mistakes in the history of physics is the continuing use of the current angular velocity and momentum equations. These equations come directly from Newton and have never been corrected. They underlie all basic mechanics, of course, but they also underlie quantum physics. This error in the angular equations is one of the foundational errors of QM and QED, and it is one of the major causes of the need for renormalization. Meaning, the equations of QED are abnormal due in large part to basic mathematical errors like this. Because they have not been corrected, they must be pushed later with more bad math: abnormal equations must be made normal.

Any high school physics book will have a section on angular motion, and it will contain the equations I will correct here. So there is nothing esoteric or mysterious about this problem. It has been sitting right out in the open for centuries.

To begin with, we are given an angular velocity ω, which is a velocity expressed in radians by the equation

ω = 2π/t

Then, we want an equation to go from linear velocity v to angular velocity ω. Since v = 2πr/t, the equation must be v = rω.

Seems very simple, but it is wrong. In the equation v = 2πr/t, the velocity is not a linear velocity. Linear velocity is linear, by the equation x/t. It is a straight-line vector. But 2πr/t curves; it is not linear. The value 2πr is the circumference of the circle, which is a curve. You cannot have a curve over a time, and then claim that the velocity is linear. The value 2πr/t is an orbital velocity, not a linear velocity.

I show elsewhere that you cannot express any kind of velocity with a curve over a time. A curve is an acceleration, by definition. An orbital velocity is not a velocity at all. It cannot be created by a single vector. It is an acceleration.

But we don’t even need to get that far into the problem here. All we have to do is notice that when we go from 2π/t to 2πr/t, we are not going from an angular velocity to a linear velocity. No, we are going from an angular velocity expressed in radians to an angular velocity expressed in meters. There is no linear element in that transform.

What does this mean for mechanics? It means you cannot assign 2πr/t to the tangential velocity. This is what all textbooks try to do. They draw the tangential velocity, and then tell us that vt = rω. But that equation is quite simply false. The value rω is the orbital velocity, and the orbital velocity is not equal to the tangential velocity.

I will be sent to the Principia, where Newton derives the equation a = v²/r. There we find the velocity assigned to the arc.¹ True, but a page earlier, he assigned the straight line AB to the tangential velocity: let the body by its innate force describe the right line AB.² A right line is a straight line, and if Newton’s motion is circular, it is at a tangent to the circle. So Newton has assigned two different velocities: a tangential velocity and an orbital velocity. According to Newton’s own equations, we are given a tangential velocity, and then we seek an orbital velocity. So the two cannot be the same. We are GIVEN the tangential velocity. If the tangential velocity is already the orbital velocity, then we don’t need a derivation: we have nothing to seek! If you study Newton’s derivation, you will see that the orbital velocity is always smaller than the tangential velocity. One number is smaller than the other. So they can’t be the same.

The problem is that those who came after Newton notated them the same. He himself understood the difference between tangential velocity and orbital velocity, but he did not express this clearly